Introduction
Geometrical optics, also known as ray optics, is a branch of optics that describes light propagation in terms of rays. This approach is valid when the wavelength of light is much smaller than the size of the structures with which it interacts. Geometrical optics is crucial for understanding how lenses, mirrors, and other optical devices work, making it an important topic for the JEE Advanced Physics syllabus.
In this study note, we'll break down the core concepts of geometrical optics, covering topics such as reflection, refraction, lens formulas, and optical instruments. Each section will include detailed explanations, equations, and examples to ensure a comprehensive understanding.
Reflection of Light
Laws of Reflection
Reflection is the change in direction of a light ray when it bounces off a surface. The laws of reflection are:
- The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
- The angle of incidence ($i$) is equal to the angle of reflection ($r$).
$$ i = r $$
ExampleExample: If a light ray strikes a plane mirror at an angle of $30^\circ$ with the normal, the angle of reflection will also be $30^\circ$.
Plane Mirrors
- Image Characteristics: The image formed by a plane mirror is virtual, erect, and of the same size as the object. The image distance ($d_i$) is equal to the object distance ($d_o$) but on the opposite side of the mirror.
- Lateral Inversion: The image is laterally inverted, meaning left and right are swapped.
To find the position of the image in a plane mirror, draw the incident and reflected rays, and extend the reflected rays backward.
Refraction of Light
Laws of Refraction (Snell's Law)
Refraction is the bending of light as it passes from one medium to another with a different refractive index ($n$). The laws of refraction are:
- The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.
- The ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) is constant and is given by Snell's Law:
$$ \frac{\sin i}{\sin r} = \frac{n_2}{n_1} $$
where $n_1$ and $n_2$ are the refractive indices of the first and second media, respectively.
Refractive Index
The refractive index ($n$) of a medium is defined as the ratio of the speed of light in vacuum ($c$) to the speed of light in the medium ($v$):
$$ n = \frac{c}{v} $$
NoteThe refractive index of vacuum is 1, and for air, it is approximately 1.
Total Internal Reflection
Total internal reflection occurs when a light ray traveling from a denser medium to a rarer medium hits the interface at an angle greater than the critical angle ($\theta_c$). The critical angle is given by:
$$ \sin \theta_c = \frac{n_2}{n_1} $$
where $n_1 > n_2$.
ExampleExample: For light traveling from water ($n_1 = 1.33$) to air ($n_2 = 1$), the critical angle is:
$$ \sin \theta_c = \frac{1}{1.33} \approx 0.75 $$ $$ \theta_c \approx \sin^{-1}(0.75) \approx 48.6^\circ $$
Common MistakeA common mistake is to apply Snell's Law incorrectly by mixing up the refractive indices of the two media.
Lenses
Lens Formula
For a thin lens, the relationship between the object distance ($u$), the image distance ($v$), and the focal length ($f$) is given by the lens formula:
$$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $$
Types of Lenses
- Convex Lens (Converging Lens): Thicker at the center than at the edges. It can form real or virtual images depending on the object's position.
- Concave Lens (Diverging Lens): Thinner at the center than at the edges. It always forms virtual, erect, and diminished images.
Magnification
The magnification ($m$) produced by a lens is given by:
$$ m = \frac{v}{u} $$
ExampleExample: A convex lens has a focal length of 10 cm. An object is placed 15 cm from the lens. Find the image distance and magnification.
Using the lens formula:
$$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $$ $$ \frac{1}{10} = \frac{1}{v} - \frac{1}{-15} $$ $$ \frac{1}{v} = \frac{1}{10} + \frac{1}{15} $$ $$ \frac{1}{v} = \frac{3 + 2}{30} = \frac{5}{30} $$ $$ v = 6 \text{ cm} $$
Magnification:
$$ m = \frac{v}{u} = \frac{6}{-15} = -0.4 $$
TipUse sign conventions carefully when applying the lens formula. For convex lenses, the focal length is positive, and for concave lenses, it is negative.
Optical Instruments
Simple Microscope
A simple microscope uses a single convex lens to magnify small objects. The magnifying power ($M$) is given by:
$$ M = 1 + \frac{D}{f} $$
where $D$ is the least distance of distinct vision (usually 25 cm) and $f$ is the focal length of the lens.
Compound Microscope
A compound microscope uses two lenses: the objective lens and the eyepiece lens. The total magnifying power ($M$) is given by:
$$ M = M_{\text{objective}} \times M_{\text{eyepiece}} $$
where $M_{\text{objective}}$ and $M_{\text{eyepiece}}$ are the magnifications of the objective and eyepiece lenses, respectively.
Telescope
A telescope is used to view distant objects. It also uses an objective lens and an eyepiece lens. The magnifying power ($M$) of an astronomical telescope is given by:
$$ M = \frac{f_{\text{objective}}}{f_{\text{eyepiece}}} $$
where $f_{\text{objective}}$ and $f_{\text{eyepiece}}$ are the focal lengths of the objective and eyepiece lenses, respectively.
NoteFor terrestrial telescopes, additional lenses are used to erect the image.
Conclusion
Geometrical optics provides a foundational understanding of how light interacts with lenses, mirrors, and other optical devices. Mastery of these concepts is essential for solving problems in JEE Advanced Physics. Be sure to practice problems and understand the principles thoroughly to excel in this topic.
By breaking down the concepts into smaller sections and using examples, this study note aims to make geometrical optics more digestible and comprehensible. Happy studying!
